Open Even Dominating Sets in Grids (even vertex v must have |N(v) intersect D| even. For a given number of columns n, we generate n by m grids having open even dominating sets. This is done by considering all 2^n-1 possible values D for the first row of the grid (the all-zero vector is not considered since it is meaningless here). In most cases, the grids are n by n (which can then be pasted together to make k(n+1)-1 by k(n+1)-1 grids having open even dominating sets for all k >= 1. However, in rarer cases, we get a value other than n. For instance, the first example below gives a 3 by 1 grid with an open even dominating set. ============================================== Enter n, the number of Columns 3 this is the array equal to count 1 0 1 0 0 0 There are 1 iterations of 1s Remainder = 3 --------------------------------- n=4 All had 4 -------------------------------- Enter n, the number of Columns 5 this is the array equal to count 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 this is the array equal to count 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 this is the array equal to count 1 0 1 0 1 0 0 0 0 0 this is the array equal to count 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 There are 1 iterations of 1s There are 3 iterations of 2s ------------------------------- n=6 All had 6 ------------------------------ Enter n, the number of Columns 7 this is the array equal to count 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 this is the array equal to count 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 this is the array equal to count 0 1 1 0 1 1 0 1 1 1 0 1 1 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 this is the array equal to count 1 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 this is the array equal to count 1 0 1 0 1 0 1 0 0 0 0 0 0 0 this is the array equal to count 1 1 0 0 0 1 1 1 1 1 0 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 this is the array equal to count 1 1 1 0 1 1 1 1 0 1 0 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 There are 1 iterations of 1s There are 6 iterations of 3s ------------------------------ Enter n, the number of Columns 8 this is the array equal to count 1 0 0 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 this is the array equal to count 0 1 0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 this is the array equal to count 1 1 0 1 1 0 1 1 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 There are 3 iterations of 2s 252 had 8 -------------------------------- Enter n, the number of Columns 9 this is the array equal to count 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 There are 1 iterations of 1s 15 itearions of 4s ------------------------------ 10 All had 10 ------------------------------ You entered the following number of Columns 11 There are 1 iterations of 1s There are 3 iterations of 2s There are 6 iterations of 3s There are 27 iterations of 5s There are 2010 iterations of 11s 10000010100 01000100010 00101000001 01000100010 10101010101 01000100010 01010001010 10001010001 11000110110 11101110111 01101100011 00101000001 01000100010 10000010100 10001010001 01010001010 10101010101 01101100011 11101110111 11000110110 11011011011 11011011011 11101110111 10101010101 11101110111 ------------------------ n=12 All had 12 ------------------------ You entered the following number of Columns 13 There are 1 iterations of 1s There are 63 iterations of 6s There are 8127 iterations of 13s 1010101010101 ------------------------- You entered the following number of Columns 14 There are 3 iterations of 2s There are 15 iterations of 4s There are 16365 iterations of 14s 10001010001010 01010001010001 01010001010001 10001010001010 11011011011011 11011011011011 -------------------------- You entered the following number of Columns 15 There are 1 iterations of 1s There are 6 iterations of 3s There are 120 iterations of 7s There are 32640 iterations of 15s 001010000010100 010001000100010 100000101000001 010001000100010 101010101010101 010001000100010 011011000110110 111011101110111 110001101100011 100000101000001 010001000100010 001010000010100 101010101010101 110001101100011 111011101110111 011011000110110 111011101110111 101010101010101 111011101110111 ----------------------- n=16 All hd 16 ----------------------- You entered the following number of Columns 17 There are 1 iterations of 1s There are 3 iterations of 2s There are 27 iterations of 5s There are 252 iterations of 8s There are 130788 iterations of 17s 01010001010001010 10001010001010001 10001010001010001 01010001010001010 10101010101010101 11011011011011011 11011011011011011 -------------------------- You entered the following number of Columns 18 n=18 All had 18 -------------------------- You entered the following number of Columns 19 There are 1 iterations of 1s There are 6 iterations of 3s There are 15 iterations of 4s There are 495 iterations of 9s There are 523770 iterations of 19s 0001010000000101000 0010001000001000100 0100000100010000010 1000000010100000001 0010001000001000100 0101010100010101010 1010001010101000101 0100000100010000010 1000001010000010100 0100010001000100010 0010100000101000001 0011011000001101100 0111011100011101110 1110001110111000111 1100000110110000011 0100000100010000010 1010001010101000101 0101010100010101010 0010001000001000100 0100010001000100010 1010101010101010101 0100010001000100010 0101010100010101010 1000000010100000001 0001010000000101000 1010001010101000101 0110001100011000110 1111011110111101111 1111011110111101111 0110001100011000110 1100011011000110110 1110111011101110111 0110110001101100011 0111011100011101110 1101010110110101011 1011011010101101101 1110001110111000111 1000000010100000001 0100000100010000010 0010001000001000100 0001010000000101000 0010100000101000001 0100010001000100010 1000001010000010100 1001010010100101001 0110001100011000110 0110001100011000110 1001010010100101001 1010001010101000101 0001010000000101000 1000000010100000001 0101010100010101010 1010101010101010101 1011011010101101101 0011011000001101100 1100000110110000011 1101010110110101011 1100000110110000011 1110001110111000111 0111011100011101110 0011011000001101100 0110110001101100011 1110111011101110111 1100011011000110110 1101010110110101011 1100000110110000011 0011011000001101100 1011011010101101101 1110001110111000111 1011011010101101101 1101010110110101011 0111011100011101110 1110111011101110111 1010101010101010101 1110111011101110111 1111011110111101111 1001010010100101001 1001010010100101001 1111011110111101111 ----------------------- You entered the following number of Columns 20 There are 3 iterations of 2s There are 63 iterations of 6s There are 1048509 iterations of 20s 10001010001010001010 01010001010001010001 01010001010001010001 10001010001010001010 11011011011011011011 11011011011011011011